Re: black hole and special relativity

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding the questions reported by Larry:

> A student asked if an object whose density is just less than than needed
> for a black hole could _look_ like a black hole to an observer in a
> reference frame moving at near the speed of light relative to the object's
> inertial reference frame, due to relativistic mass increase.  Could it be a
> black hole to some observers and not to others?

No, and no.  A black hole is a black hole in all external reference
frames.  Also, it is *not* possible to have an object whose (presumably
average) density is just a little less that that needed for it to have
an event horizon just outside it so it could still be a stable
configuration just shy of being a black hole.  It is possible to prove a
theorem that says that if a spherically symmetric object is stable
against gravitational collapse then its radius *must* be strictly greater
than 9/8 of the Schwarszchild radius for that mass if it was collapsed to
a (Schwarszchild) black hole.  Thus, it is not possible to have a stable
object whose radius is just a little larger than what its black hole
horizon radius would be if it was a black hole.

> Another student asked if General Relativity is a superset of Special
> Relativity; i.e., can all the results of SR be derived as special cases
> from GR?

Yes and no, the result depending on how you interpret the phrases
"all the results", and "special cases".  It is true that in GR if one
looks at a sufficiently small neighborhood of any spacetime event, then
in that region all the results of SR hold to any specified accuracy.
The relevant practical criterion is for the various curvature lengths for
various directions in spacetime to be enormous compared to the size of
the spacetime region along each of those directions.  But if a given
global region of spacetime has a finite curvature in it then the
predictions of GR will disagree with those of SR over that region by some
finite amount.  SR is a special case of GR in the sense that in a region
of spacetime devoid of any true gravitational effects then SR will
correctly hold in that region.  But for a region of spacetime containing
gravitational sources or gravitational waves (or both), then SR is *just
wrong* for that region.

> After all, aren't inertial reference frames just special cases of
> accelerated reference frames (where a = 0)?

True.  But whether or not a frame is accelerated or not has nothing
necessarily to do with whether GR is necessary or not.  The criterion
for the necessity of GR is the presence or absense of spacetime
curvature, not the presence or absense of an inertial coordinate system.
It is true, though, that an intrisically curved spacetime will not
possess an inertial coordinate system in terms of Cartesian coordinates
that is valid everywhere in that spacetime.  For such a system, even if
one part of spacetime is in an inertial frame, some other part is not
for the same coordinate system.

David Bowman
David_Bowman@georgetowncollege.edu